5 Hirsch-Fye quantum Monte Carlo method for ... - komet 337

Hirsch-Fye QMC 5.25
(a) 0.0236
(b)
D
0.0235
0.0234
0.0233
0.0232
0.0231
0.023
0.0229
0.0228
0 10 20 30 40 50 60 70
it
∆τ=0.12
∆τ=0.15
∆τ=0.18
∆τ=0.20
D
0.024
0.023
0.022
0 0.02 0.04
∆τ
0.06
2
Fig. 15: Steps in the HF-QMC based computation of the double occupancy atU = 5,T = 0.04:
(a) estimates can be calculated as averages with statistical error bars from the analysis of time
series (traces), taking autocorrelation into account, for each value of∆τ. (b) Numerically exact
results are obtained in a second step using extrapolation by least-squares fits.
be included in averages. Moreover, all curves show significant autocorrelation which has to
be taken into account for error analysis. Taking these issues into account, standard analysis
techniques for time series yield raw HF-QMC results for each value of the discretization plus
an error bar which takes statistical and convergency error into account. Such data is shown in
Fig. 15b as a function of the squared discretization. Evidently, the discretization dependence is
very regular; a straightforward extrapolation using standard least-square fit methods (here with
the 3 free parameters corresponding to the orders ∆τ 0 , ∆τ 2 , and ∆τ 4 ) essentially eliminates
this systematic error, i.e. reduces the total error by two orders of magnitude. 19
Due to the cubic scaling of the effort with the number of time slices, the total cost of achieving
extrapolated results for some grid range is dominated by the smallest discretization. Thus the
possibility of extrapolation comes (with the use also of coarser grids) at no significant cost; in
contrast, DMFT convergence can be accelerated by using the ∆τ hysteresis technique outlined
above. Taking all of this into account, HF-QMC with extrapolation can be competitive with
the recently developed continuous-time QMC solvers, as demonstrated in Fig. 16: at fixed total
computer time, this method achieves the highest precision [47] in energy estimates for a test
case established in [43].
Note that the use of insufficiently converged solutions is potentially a very significant source of
errors. It is important to realize that in principle measurements have to be performed exactly at
the solution of the self-consistency equations, i.e., for the exact bath Green function. Averages
over measurements performed for different impurity models corresponding to approximate solutions
do not necessarily converge to the exact answer in the limit of an infinite number of
models (i.e., iterations) and measurements. 20 Still, the most important practical point when
19 The reader may note a similarity to the MC example shown in Fig. 3.
20 Trivially, a measurement of the free energy F itself (using a suitable impurity solver) is a good example.
Since F is minimal for the true solution, all measurements taken for approximate solutions will be too large. The
correct answer can, therefore, not be approached by averaging over many measurements, but only by reducing the